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Introduction to Statistics
• Exercises can be solved in any order.
• Most exercises are made up of a few independent items, thus, if you are not able to solve one of them, do not skip the whole exercise but read and try the next item.
1. In a study of brand recognition, 831 consumers knew of Campbell’s Soup, and 18 did not (based on data from Total Research Corporation). Use these results to estimate the probability that a randomly selected consumer will recognize Campbell’s Soup. [3 points]
2. A snack cracker company conducted a taste test for the three different types of crackers it makes. It surveyed 250 people in each age group in the table below. Participants chose their favourite type of cracker. Use the results to answer the questions.
Age Cracker A Cracker B Cracker C Under 20 152 54 44 20 to 39 107 85 58 40 to 59 78 101 71 60 and over 34 68 148
(a) What is the probability that a participant was over 60 years old and chose Cracker C? [1 point]
(b) What is the probability that a participant did not choose Cracker A? [2 point]
(c) What is the probability that a participant chose Cracker A or was under 20 years old? [3 point]
(d) A randomly selected participant says he is 15 years old. What is the probability that he chose Cracker B? [2 points]
3. A psychology professor gives a surprise quiz consisting of 10 questions. Each questions has 3 choices, one of which is correct. The professor states that passing requires at least 8 correct responses. Assume that an unprepared student adopts the questionable strategy of guessing for each answer.
(a) Find the probability that the first 8 responses are correct (and the last 2 are wrong). [2 points]
(b) Is the probability from part (a) equal to the probability of passing? Why or why not? [2 points]
4. A marble is randomly chosen from a can containing 4 red and 3 blue marbles. It is replaced by two marbles from the other colour. Another marble is then randomly chosen from the can.
(a) Determine the probability that the marbles chosen have the same colour! (Hint: using a tree diagram can be useful) [5 points]
(b) Given that the marbles have the same colour what is the probability that they are both blue? [3 points]
5. When a pair of dice is rolled, the random variable M denotes the larger of the two numbers that are shown uppermost, or the value of a single die if a double is thrown.
(a) Write down and draw the probability distribution of the random variable M . [6 points]
(b) Calculate the expected value and the variance of the distribution. [5 points]
(c) Find P (M ≤ 2). [2 points]
6. An association of coffee producers has estimated that 60% of adult workers drink coffee at least occasionally while on the job. Given this estimate, for a randomly selected group of 12 workers:
(a) What is the probability that no more than 3 of them drink coffee at least occasionally on the job? [4 points]
(b) What is the expected number of workers drink coffee at least occasionally on the job? [1 point]
7. The first aid tent at a music festival can deal with no more than three people requiring treatment in any one hour. The mean number of people requiring treatment is two per hour.
(a) Is this binomial or Poisson distribution? What is the expected number of people requiring treatment in an hour? [2 points]
(b) What is the probability that exactly 1 person will require treatment in an hour? [3 points]
(c) What is the probability that they will not be able to deal with all the people requiring treatment in an hour? [5 points]
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